BOSTON — Jose Abreu has enjoyed his first look at Fenway Park, the historic venue that very nearly was his home.

While the first-time All-Star said Tuesday he left the free agency process that brought him to the White Sox to his agent, Barry Praver, multiple sources said the Boston Red Sox were a very serious player for the first baseman’s services.

Ultimately, Abreu signed a six-year, $68-million deal with the White Sox, who have begun to emerge from the depths of a 99-loss season in part because of how the slugger has brought vitality to an offense that failed to score 600 runs in 2013.

White Sox general manager Rick Hahn told the Boston Herald on Monday he knew of the Red Sox interest in Abreu, an attraction that was in part kept out of the spotlight because of Boston’s postseason run.

Given how Abreu has energized the franchise, Hahn doesn’t know how the White Sox offseason would have gone had the Cuban free agent wound up in Boston, which reportedly offered only $5 million less.

“It’s tough to know exactly what would have followed had we not been able to convert on that deal,” Hahn said on Tuesday. “But it certainly addressed a very important hole for us and plugged a need for the next several years and allowed us to sort of move on to some other (moves). Whether we would have been able to plug that hole through other moves, hard to say. But certainly its pretty safe to say it wouldn’t have been as strongly as we were able to by getting the deal done.”

An All-Star in his first season, Abreu’s arrival was significant for several reasons. First and foremost, the White Sox added a middle-of-the-order bat at the start of his physical prime, a player Paul Konerko has described as a rookie without being a rookie.

Given this is Abreu’s first go-round through the league, the White Sox feel as if they’ve tapped into a gold mine.

Even though he’s seeing many pitchers for the first time and acclimating to life in a new culture and a new league, Abreu has already clubbed 27 home runs. He also has become a leader by example in the clubhouse and the team believes there’s more growth potential as Abreu learns English.

Then there’s the other aspect — Abreu came at only the cost of money.

The White Sox didn’t have to surrender any draft picks to sign Abreu because no team held his rights, nor did they have to use any of their own players to acquire him in a trade. That proved valuable in December when the White Sox traded Addison Reed for third-base prospect Matt Davidson and Hector Santiago for leadoff man Adam Eaton.

Hahn likes to joke that his selling point to White Sox chairman Jerry Reinsdorf on the Abreu deal was ‘Its just money.’ ”

“But there’s some truth to that,” Hahn said. “Obviously it was very real money and would put a significant dent into our payroll, especially if it hadn’t gone right. But ultimately it didn’t cost us any other assets. That was a big part of the reason we were willing to take that risk, because it was just cash.”

In need of revitalizing an aging offensive core, the White Sox liked Abreu so much they gave him an international-record deal at the time. Some early estimates suggested Abreu would fetch in the $40 million range. But Praver said in November five teams crested $60 million in pursuit of Abreu.

Boston was one of the closest, Hahn told the Herald.

“I certainly felt there were numerous teams, any of whom could jump up and jump over us or push us to a higher level,” Hahn told the Herald. “After the fact, we got a better sense of who was there and where they were going, and it turns out there were several teams, not just us and Boston, that were awfully aggressive and pretty close to where we were at the end.”

Asked how close the Red Sox were to signing him, Abreu said through Lino Diaz, the team’s manager of cultural development, that he didn’t want to know. But he would agree that he is impressed with Fenway and it’s storied past.

“I am aware of the history and to be honest with you it is an honor to play here,” Abreu said. “And given the age and everything, it’s a place that has a great environment to play in. It’s an honor to play here for the first time.”

Hahn doesn’t want to think about any what ifs.

He and the White Sox are ecstatic to have Abreu on board. That, and he has already seen a preview of what 2014 might have looked like with Abreu and wasn’t happy with the image.

“I didn’t like how it looked those two weeks he was on the (disabled list),” Hahn said. “So hopefully had we not signed him we would have found another way to try to plug that hole and it would have looked a little different than it (has). But obviously he’s a huge cog in that lineup and hopefully will be for the next five-plus seasons.”